<h2>Problem 273</h2>
<div style="color:#666;font-size:80%;">09 January 2010</div><br />
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<p>Consider equations of the form: <var>a</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <var>b</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = <var>N</var>, 0 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>a</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>b</var>, <var>a</var>, <var>b</var> and <var>N</var> integer.</p>

<p>For <var>N</var>=65 there are two solutions:</p>
<p><var>a</var>=1, <var>b</var>=8 and <var>a</var>=4, <var>b</var>=7.</p>
<p>We call S(<var>N</var>) the sum of the values of <var>a</var> of all solutions of <var>a</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <var>b</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = <var>N</var>, 0 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>a</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>b</var>, <var>a</var>, <var>b</var> and <var>N</var> integer.</p>
<p>Thus S(65)=1+4=5.</p>
<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' />S(<var>N</var>), for all squarefree <var>N</var> only divisible by primes of the form 4<var>k</var>+1 with 4<var>k</var>+1 <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 150.</p>

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